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# Differential Equations for Applications

Math 3410, Sections 001 and 002 - Elementary Differential Equations, Fall 2017 [Course Syllabus]

**Lectures:** MWF 09:05 - 09:55 at LH 201.

**Office hours:** Monday 12:00 - 13:30, Wednesday 12:30 - 14:00 at MONT 304.

### Lecture Notes

- Note 1 -- Note 2 -- Note 3 -- Note 4 -- Note 5(We skipped the applications and we will come back to that later).
- Note 6 is about Autonomous Equations and Stability of Equilibrium Solutions.
- Current Lecture Note --> Note 7 is about Secon Order Linear equations and Laplace Transform.
### Quizzes

- Solution to Quiz 1.
### Homework

#### HW1 - Due on Friday, September 8 by the class | Solution to selected problems (PDF)

- Problem 1: Show that $e^{2x}+e^{2y}=1$ is an implicit solution to the DE $e^{x-y}+e^{y-x} \frac{dy}{dx}=0$.
- Problem 2: Find a 1-parameter family of solutions of the DE $y'=y$ and the particular solution for which $y(3)=1$.
- Problem 3: Construct a direction field for the differential equation $y'=2x$.
- Problem 4: Find a particular solution to the DE $y'=e^{x+y}$ with the initial value $y(0)=0$.
#### HW2 - Due on Friday, September 15 by the class | Solution to selected problems (PDF)

- Problem 1: Check that if the differential equation is exact $(e^{x}\sin y+e^{-y})dx - (xe^{-y} - e^{x} \cos y) dy= 0$. If it is exact then solve the differential equation.
- Problem 2: Check that if the differential equation is exact $e^{x}(x+1) dx +(ye^y - xe^x)dy= 0$. If it is exact then solve the differential equation.
- Problem 3: Let $P(x)=\int p(x) dx$. Show that $e^{P(X)}$ is an integrating factor for the DE

\[ y'+p(x) y=q(x). \] - Problem 4: Suppose that $a,b,c,e$ are constants such that $ae-bc\neq 0$. Let $m$ and $n$ be arbitrary real numbers. Show that
\[
(ax^{m}y+by^{n+1})dx + (cx^{m+1}+exy^{n})dy=0
\]
has an integrating factor $\mu(x,y)=x^{\alpha}y^{\beta}$ for some $\alpha$ and $\beta$.

#### HW3 - Due on

~~Friday, September 22~~Monday, September 25 by the class- Problem 1: Find the orthogonal trajectories of the family of circles centered on $x$-axis and passing through the origin.
- Problem 2: Find the orthogonal trajectories of the family of curves having equation $e^x \cos(y)=k$.
- Problem 3: Find the general solution of the Bernoulli equation $xy'+y+x^2y^2e^x=0$.(You may need to rewrite the equation!).
- Problem 4: Find the general solution of the Bernoulli equation $x^2y'+2y=2e^{\frac{1}{x}}y^{\frac{1}{2}}$.
- Problem 5: Find the general solution of the Ricatti equation $y'=1+\frac{y}{x}-\frac{y^2}{x^2}$ with given particular solution $y_1(x)=x$.
- Problem 6: Find the general solution of the Ricatti equation $y'=y^2+2xy+(x^2-1)$ with given particular solution $y_1(x)=-x$.
### Announcements and Grades

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